qstn 1

Consider the following joint probability density function of the random variable X and Y: f(x) = {48/x^4 , x>2,0

1. a. Find the marginal distribution for X and Y, remember to write the range of X and Y such that fx(x), fy(y)is positive.
2. b. Determine whether X and Y are independent.
3. c. Find the expected value of Y and X.
4. d. Find the variance of Y and X.
5. e. Find the covariance of X and Y, Cov(X,Y)

f. Find the correlation coefficient xy

qstn 2

There are 15 different marbles and 3 jars. Suppose you are throwing the marbles in the jars and there is a 20%, 50% and 30% chance of landing a marble in jars 1, 2 and 3, respectively.

1. a. What is the probability of landing 4, 6 and 5 marbles in jars 1, 2 and 3 respectively?
2. b. Suppose that out of the 15 marbles 7 are red and 8 are blue. If we randomly select a sample of size 5, what is the probability that we will have 3 blue marbles?
3. c. Suppose we will throw marbles at the jars, until we have landed three (regardless of color) in jar 1. What is the probability that we will need to throw ten marbles to achieve this?

qstn 3

Hospital administrators in large cities anguish about traffic in emergency rooms. At a particular hospital in a large city, the staff on hand cannot accommodate the patient traffic if there are more than 10 emergency cases in a given hour. It is assumed that patient arrival follows a Poisson process, and historical data suggest that, on the average, 5 emergencies arrive per hour.

1. a. What is the probability that in a given hour the staff cannot accommodate the patient traffic?
2. b. What is the probability that more than 20 emergencies arrive during a 3-hour shift?
3. c. Write down the name and the parameters of the distribution that describe the time until the first patient arrives.
4. d. Write down the name and the parameters of the distribution that describe the time until the first two patient arrives.